Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Definition of a Sequence: A list of numbers that follows a specific rule or pattern.
Term: Each individual number within a sequence.
Term-to-term rule: A rule that describes how to get from one term to the next (e.g., 'Add 3').
Common Difference: The constant value added or subtracted to get the next term in an arithmetic sequence.
Arithmetic Sequence: A sequence where the difference between consecutive terms is always the same.
Position-to-term rule (n-th term): A rule that allows you to calculate any term in a sequence based on its position (n) in the list.
Geometric Sequence: A sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Special Sequences: Recognizing Square numbers (1, 4, 9...), Cube numbers (1, 8, 27...), and Fibonacci sequences (1, 1, 2, 3, 5...).
📐Formulae
Common Difference:
General form of a linear -th term:
Finding the -th term of an arithmetic sequence:
Square numbers sequence:
💡Examples
Problem 1:
Find the next two terms and the term-to-term rule for the sequence: 7, 13, 19, 25, ...
Solution:
Terms: 31, 37. Rule: Add 6.
Explanation:
Calculate the difference between consecutive terms: and . Since the difference is constant, add 6 to the last known term (25) to get 31, and add 6 to 31 to get 37.
Problem 2:
Find the -th term formula for the sequence: 5, 8, 11, 14, ...
Solution:
Explanation:
- Find the common difference (): . This means the formula starts with . 2. Test : . To get the first term (5), we need to add 2. Therefore, the formula is .
Problem 3:
What is the 10th term of a sequence with the -th term formula ?
Solution:
105
Explanation:
Substitute into the formula: .